Plane Geometry - Bruce E. Shapiro

SECTION 44. ESTIMATING π 245ThuswhereThis converges very quicklyÅ Å ã Å ãã1 1π = lim 4 4fn→∞ 5 , n − f239 , nf(x, n) =n∑k=0x 2k+1 (−1) k2k + 1nπ n0 3.18326359832635983261 3.14059702932606031432 3.14162102932503442503 3.14159177218217729504 3.14159268240439951725 3.14159265261530860816 3.14159265362355476207 3.14159265358860222878 3.14159265358983584759 3.141592653589791696910 3.1415926535897932947Another version of this method is to sum the terms in equation 44.1 usingan Euler transform. A version of this known as Van Wijngararden Transformation,was invented in the 1960’s, tells us that we do not have to sumthe whole series to get a better approximation. 4 .Theorem 44.2 (Euler Transform) Let a k be a converging sequence ofpositive numbers, such that the partial sumss k,0 =k∑(−1) n a nn=0coverge to some number S. Then the sequence s k,j+1 → S, wheres k,j+1 = s k,j + s k+1,j2Setting π k,0 = f(x, k) for all k, then computing the partial sums as π k,j+1 =(π k,j + π k+1,j )/2, we can compute approximations of π. Van Wijngaardendeveloped an efficient algorithm for computing the sum without having torecompute earlier values in the sequence of partial sums. Table 44.1 showsthe number of digits accuracy as indicated by log 10 |(π i,j − π)/π|.4 A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Process Analyse, StichtingMathematisch Centrum, (Amsterdam, 1965) pp. 51-60.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.